q-theta function

In mathematics, the q-theta function (or modified Jacobi theta function) is a type of q-series which is used to define elliptic hypergeometric series. It is given by

[math]\displaystyle{ \theta(z;q):=\prod_{n=0}^\infty (1-q^nz)\left(1-q^{n+1}/z\right) }[/math]

where one takes 0 ≤ |q| < 1. It obeys the identities

[math]\displaystyle{ \theta(z;q)=\theta\left(\frac{q}{z};q\right)=-z\theta\left(\frac{1}{z};q\right). }[/math]

It may also be expressed as:

[math]\displaystyle{ \theta(z;q)=(z;q)_\infty (q/z;q)_\infty }[/math]

where [math]\displaystyle{ (\cdot \cdot )_\infty }[/math] is the q-Pochhammer symbol.

See also

• elliptic hypergeometric series
• Jacobi theta function
• Ramanujan theta function